Quantitative Illumination of Convex Bodies and Vertex Degrees of Geometric Steiner Minimal Trees

Two results are proved involving the quantitative illumination parameter B ( d ) of the unit ball of a d ‐dimensional normed space introduced by Bezdek (1992). The first is that B ( d ) = O (2 d d 2 log d ). The second involves Steiner minimal trees. Let v ( d ) be the maximum degree of a vertex, and s ( d ) that of a Steiner point, in a Steiner minimal tree in a d ‐dimensional normed space, where both maxima are over all norms. Morgan (1992) conjectured that s ( d ) ≤ 2 d , and Cieslik (1990) conjectured that v ( d ) ≤ 2(2 d − 1). It is proved that s ( d ) ≤ v ( d ) ≤ B ( d ) which, combined with the above estimate of B ( d ), improves the previously best known upper bound v ( d ) < 3 d .